Chapter 1.8, Problem 81E

To determine

To calculate:Whether the equation $y=x^3+10x$ is symmetric about x-axis or y-axis or origin.

Answer to Problem 81E

The equation $y=x^3+10x$ is symmetric only with respect to origin.

Explanation of Solution

Given information:

The equation $y=x^3+10x$ .

Formula used:

The function is symmetric about the x-axis, when y is replaced by $-y$ , the equation remains unchanged.

The function is symmetric about the y-axis, when x is replaced by $-x$ , the equation remains unchanged.

The function is symmetric with respect to origin, when y is replaced by $-y$ and x is replaced by $-x$ , the equation remains unchanged.

Calculation:

It is provided that the equation is $y=x^3+10x$ . Construct a table to evaluate the value of y for different values of x.

Recall that the function is symmetric about the x-axis, when y is replaced by $-y$ , the equation remains unchanged.

Replace y by $-y$ in the equation $y=x^3+10x$ ,

  $-y=x^3+10x$

The equation is changed. Therefore, the equation $y=x^3+10x$ is not symmetricabout the x-axis.

Recall that the function is symmetric about the y-axis, when x is replaced by $-x$ , the equation remains unchanged.

Replace x by $-x$ in the equation $y=x^3+10x$ ,

  $\begin{array}{c}y={\left(-x\right)}^3+10\left(-x\right)\\ y=-x^3-10x\end{array}$

The equation is changed. Therefore, the equation $y=x^3+10x$ is not symmetricabout the y-axis.

Recall that the function is symmetric with respect to origin, when y is replaced by $-y$ and x is replaced by $-x$ , the equation remains unchanged.

Replace x by $-x$ and y by $-y$ in the equation $y=x^3+10x$ ,

  $\begin{array}{c}-y={\left(-x\right)}^3+10\left(-x\right)\\ -y=-x^3-10x\\ y=x^3+10x\end{array}$

The equation is unchanged. Therefore, the equation $y=x^3+10x$ is symmetricabout the origin.

Thus, the equation is symmetric only about origin.